SINGULARLY
IMPORTANT
IMPORTANT
The SVD of A produces three matrices: two of this special kind, as well as one of this other kind.
What are "two unitary matrices and one diagonal matrix"?
If "U" don't want the error to blow up, use this option when solving Ax = b "a-la Gauss"
What is "row pivoting"?
This value represents the difference between 1 and the floating point value closest to it, at least on your hardware
What is "machine epsilon"?
Let's get into shape: reduce A to this form before racing off to find eigenvalues
What is "(upper) Hessenburg"?
|x^Ty| leq ||x||*||y||
What is the "Cauchy-Schwarz inequality"?
I added up the eigenvalues of A, and all I got was this value, which is itself defined as a different sum altogether!
What is the "trace"?
Keep tally of these scalars if you want to compute rank(A) = rank(U Sigma V^T)
What are "nonzero singular values"?
Don't F-A-I-L to count these up when predicting the runtime of your algorithm!
What are "FLOPS"?
It takes this many bits to represent a single complex number in double-precision floating point
What is "128"?
This one "bullies" the others into submission when you apply the Power Method
What is the "dominant eigenvalue"?
(||x||*||J(x)||)/||f(x)||
This quantity is defined as sqrt(tr(A^**A)), although it would be unwise to actually compute it that way!
What is the "Frobenius norm"?
The singular values of A in bbb C^n are the square roots of the eigenvalues of this related matrix
What is "A-star-A"?
Give the matrix a little "spin" by using this method to ortho-normalize its column vectors
What is "Givens rotations"?
The "laws" of floating point are laid down in a standard which bears this alphanumeric name
What is "IEEE 754"?
If A = QR, then this term refers to the relation which guarantees that A and RQ have the same eigenvalues
What is "(unitary) similarity"?
max_x ||Ax||/||x||
What is the "operator norm"?
If you want to fit a polynomial to data (x_i,y_i), you can use the matrix bearing this "V" name. Just fill each row with [1,x_i,x_i^2,ldots,x_i^n]
What is a "Vandermonde matrix"?
This SVD-derived ratio is another way of computing the matrix condition number
What is the "the biggest singular value divided by the smallest"?
Named for a titan of numerical analysis, this adaptive adjustment can help you find eigenvalues in a hurry
What is the "Wilkinson shift"?
Floating-point addition lacks this familiar "order-ignoring" property of regular addition. You might say the two don't really get along!
What is "associativity"?
We're pretty confident that QR iteration converges to this useful factorization of A
What is the "Schur decomposition"?
||x- hat x||/||x||
when f_A(x) = f(hat x)
What is "backward error"?
A creative "remix" of U_r, Sigma_r, and V_r gives you this least-squares-solving matrix
What is the "(Moore-Penrose) pseudoinverse"?
Written I - 2u u^T, these transformations can help "bring home" the QR factorization
What are "Householder reflections"?
This term, which refers to the "significant digits" of a floating-point number, might also bring to mind a kind of insect
What is "the mantissa"?
This sad-sounding technique actually saves you time when you're trying to finish finding the eigenvalues of A
What is "deflation"?
max_j sum_i |a_(ij)|
What is the "matrix 1-norm"?